Final Results: Effects of Sex and Mixing on Male-Bias of TB

Description of Simulation Study

In this simulation study of pathogens spreading on contact networks, we sex-assortativity (r=0 to 0.6 by 0.2) and sex-trait ratios (\(\alpha\)) to see how the ratio of male-female cases changes.

Assorted networks were generated in two ways: with methods in Sah et al. (2019) and with a re-wiring algorithm which selectively re-wires same-sex edges until the desired level of assortativity is reached (within a small value, \(\epsilon=0.05\)). All networks had mean degree of 10.

We investigated the following sex-traits: susceptibility (SUS), infectious period (IP), and transmissibility (TRA). We investigated different strengths of these sex-traits (\(\alpha\)).

Rates of susceptibility for nodes (SUS) was modeled as

Source -> Target	Overall transmission rate, \(\beta\)
F->F	\(\frac{2 \tau } {(\alpha + 1)}\)
M->M	\(\frac{2 \tau \alpha} {(\alpha + 1)}\)
M->F	\(\frac{2\tau } {(\alpha + 1)}\)
F->M	\(\frac{2 \tau \alpha} {(\alpha + 1)}\)

Rates of transmissibility (TRA) for nodes was modeled as

Source -> Target	Overall transmission rate, \(\beta\)
F->F	\(\frac{2 \tau } {(\alpha + 1)}\)
M->M	\(\frac{2 \tau \alpha} {(\alpha + 1)}\)
M->F	\(\frac{ 2 \tau \alpha} {(\alpha + 1)}\)
F->M	\(\frac{2 \tau} {(\alpha + 1)}\)

Infectious period (INF) for nodes modeled as

Source	Overall transmission rate, \(\beta\)
F	\(\frac{\gamma (\alpha + 1) } {2}\)
M	\(\frac{\gamma (\alpha + 1) } {2\alpha}\)

Each sex-trait was modeled separately.

Sensitivity parameters tested:

1. transmission rate (\(\tau=0.04, 0.075, 0.1\), \(R_0=1.5, 2.5, 3.5\))
2. model type (SIR, SLIR, SIRS, SLIRS). SIRS and SLIRS models ran for 200 time units.
3. Network type: SW (rewired), SF (rewired), Geometric (Sah Algorithm)

Response variables:

1. Male-bias: calculated differently for SIR/SLIR and SIRS/SLIRS. For models without recovery: number of male recovered nodes at end of simulation divided by number of female recovered nodes at end of simulation. For models with recovery: average ratio of male cases to female cases in last 100 time units for each simulation.
2. Epidemic duration: calculated for SIR/SLIR models as the number of time units before infectious population reached 0.
3. Total number infected: calculated for SIR/SLIR models as the total number of individuals that became infected before infectious population reached 0.
4. Prevalence of latent infection: calculated for SLIRS model as the size of the latent population at the last time step

Structure of Assorted Networks

Plots of networks generated with re-wiring algorithm, shown is a non-assorted network compared with the most assorative network tested here. Node color represents different sexes (modules).

We see from the plot below that, with the exception of the Sah algorithm, important network statistics change as assortativity increases. This is not surprising given that the Sah algorithm is designed to hold other parameters constant. This is something to keep in mind as other results are interpreted.

Changes in network statistics using the re-wiring algorithm to make networks of varying assortativity.

Model parameters: \(R_0\)

Below we examine the relationship between the epidemic threshold (\(R_0=1\)) as calculated in Kiss, Miller, & Simon (2017) and simulated epidemics on networks with different values of \(\tau\). Other parameters are given in figure caption. The epidemic threshold is given by:

\(\frac{\tau <K^2-K>}{\tau+\gamma <K>}>1\)

Comparison between analytically calculated \(R_0\) (Kiss, Miller, Simon 2017) and simulated epidemic size on non-assorted and assorted networks. Horizontal grey line shows where \(R_0=1\) and where ending epidemic size > 0. Vertical grey line approximately where \(R_0=1\). Results are consistent for SIR and SLIR models. No sex-trait heterogeneity is included in these simulated data. Other parameters: $\(\gamma=0.5; I_0=.01; \psi=0, 0.1; \delta=0.25, 1\cdot6\)

We see that the threshold matches up with simulated results (i.e., where \(R_0=1\)) epidemics are possible (i.e., outbreak size > \(I_0\)).

For \(\tau\) parameters used in this study, mean \(R_0\) on SAH networks for SLIR models is given below.

Tau values and estimated R0 for SLIR models simulated on SAH networks.

tau	mean_r0
0.040	1.504810
0.075	2.452056
0.100	3.374769

Model parameters: Equilibrium of latent infections and sex-bias

While the problem investigated with these models is inspired by TB, the models themselves are very simple, and not parameterized to any specific dataset. A marked trait of TB is that, globally, 25% of the world’s population has latent infection. Here, we investigate the equilibrium latent infections and case ratio for different infection rates and strengths of sex traits. We think these data points give an idea of the parameters that we should be focusing on with these models specifically for investigating male-bias in TB.

Latent equilibrium and case-ratio on Sah networks.

Only Sah results are shown here but conclusions from simulations on rewired networks were similar.

It looks like \(\tau=0.075\) or \(\tau=0.1\) results in an equilibrium latent prevalence of 25%. In the next section, we look at results with \(\tau=0.075\) and then we consider the results with a slower spreading pathogen \(\tau=0.04\) and faster spreading pathogen \(\tau=0.1\).

Assortativity, Sex Traits, & Male Bias in SIR, SLIR, SIRS, and SLIRS models

Best guess for \(\tau\) (\(\tau=0.075, R_0=2.5\))

First, we show results for simulated male-bias with \(\tau=.075\) on SAH networks with varying \(\alpha\).

Results for the Sah networks above are representative of male-bias/assortativity results from other networks:

• Assortativity alone (where \(\alpha=1\) (no difference in sex trait)), even in very assorted networks, can not lead to male-bias in cases
• Higher male susceptibility and longer male infectious periods can lead to male bias without assortativity in some models but higher transmissibility can’t lead to male bias without assortativity
• Assortativity increases male bias when \(\alpha > 1\)
• In SIR and SLIR model, only higher male susceptibility can lead to male bias without assortativity
• Pathogens that do not have immunizing infections (SIRS, SLIRS) result in much more variable amounts of sex-bias in infections whereas SIR and SLIR result in lower variation in sex-bias
• Higher susceptibility seems to lead to more male-bias than other sex-traits, given a certain model type (e.g., for SLIR)

Results are similar on re-wired networks (only SLIR results shown below).

The graph above, which compares simulations on different network types, shows how higher male susceptibility is the only sex trait that can lead to male bias with assortativity.

With some assortativity, male bias increases with longer male infectious periods and higher male transmissibility but doesn’t seem to reach global levels of male-bias in cases.

How rate of transmission affects results

Here we show the sensitivity of these results to different \(\tau\) values for SLIR model.

And here we show the sensitivity of these results to different \(\tau\) values for SLIRS model.

Faster spreading pathogens typically result in lower male-bias than slower spreading pathogens.

Parameter combinations leading to male-bias

The heat plots below show which parameter combinations lead to male:female case ratios greater than 1.25 given our best guess \(\tau\) estimate of 0.075.

The effects of sex-traits and assortativity on male-bias in simulations on SAH-networks. Results shown for low transmissability pathogen. Only ratios above 1.25 are shown to visualize which parameter combinations in which models can lead to male-bias.

The above plot shows possible combinations that lead to male-bias. Any combinations without color lead to male:female case ratios < 1.25.

Epidemic dynamics on assorted networks

SIR & SLIR

The mean peak, duration, and final size does not vary much (heat plots not shown) with assortativity. To see the distribution of these values and results on different networks, see below box plots which are colored by assortativity values throughout this analysis.

The simulations on Sah networks clearly show that assortativity does not affect peak size, final size, or duration of epidemic. Assortativity within rewired networks, with their changing structures (e.g., decreased clustering in assorted SW networks), does seem to alter epidemic dynamics. However, it’s most likely due to the rewired algorithm, especially since these results aren’t reproduced on Sah networks.

Focusing on outbreaks on Sah networks, below we see whether differences in sex-traits impact epidemic dynamics.

In the plot above, SIR results are shown on top and SLIR results on bottom.

It seems infectious period and susceptibility can alter epidemic dynamics more than transmissibility.

Specifically, higher male susceptibility can lead to smaller overall outbreaks (SIR and SLIR). Peak size and duration was not noticeably changed by higher male susceptibility.

Also, longer male infectious periods can lead to longer, lower peaked SIR epidemics (less so for SLIR epidemics). Longer infectious periods did not have a large effect on final outbreak size.

Transmissible did not appreciably change peak size, final size, or duration.

SLIRS

Finally, in SLIRS model, how does assortativity change equilibrium latent prevalence?

Increased male susceptibility relative to female susceptibility can decrease latent equilibrium infections.

List of Conclusions

• Assorted networks that were made with our “rewiring” algorithm have changes in clustering, path length, and degree assortativity that can alter how diseases spread. The Sah algorithm was designed to keep these values constant, and our analyses are consistent with their claims.
• The epidemic threshold, where \(R_0=1\) can be calculated using results from Miller, Kiss, and Simon and their equation matches up with our simulated outbreaks.
• These simulations are not fitted to any data sources, and modeling TB is subject to many assumptions that we did not want to make. However, the observation of male-bias in cases is inspired by TB case reports and we wanted our model to match up with some key characteristics seen globally. Specifically, we wanted to match modeled equilibrium latent prevalence with global data which indicates approximately 25% of the world’s population has latent infection. Additionally, we wanted to match male:female cases to the globally reported ratio which is around 1.8. Using these two data points, we think \(\tau \elem (0.075, 0.1)\).
• Focusing on Sah networks for the analyses, we find assortativity alone can’t lead to male-bias (even in highly assorted networks)
• While, higher male susceptibility and longer male infectious periods can lead to male bias without assortativity in some model, higher transmissibility can’t lead to male bias without assortative mixing
• Additionally, assortativity amplifies male-bias in the presence of sex-differences in infectious period, susceptibility, and transmissibility
• Higher susceptibility seems to lead to more male-bias than other sex-traits, given a certain model type (e.g., for SLIR)
• Faster spreading pathogens are less likely to be affected by differences in sex-traits or assortativity. The contact network characteristics and sex-differences in infection rates are less important than the spread of infection.
• Epidemic dynamics are generally not affected by assortativity (according to Sah networks). Simulations on rewired networks, which had inherent changes in clustering etc., did show some relationships with assortativity and duration, peak size, etc. But we believe these results are due to changes in other network structures (not assortativity itself).
• Interestingly, higher male susceptibility can decrease outbreak size in SIR and SLIR networks (but limited relationship to assortativity).

Take home

• Assortativity, combined with sex-traits, can amplify ratios of male:female infections
• At realistic levels of assortativity (0.2-0.3), …